Lozenge Tilings · affine·mess

Flat view. Click a lozenge to turn it, drag to move it.

Three colours, equal shares

Tile a regular hexagon of side n with lozenges, each made of two unit triangles. A lozenge can point three ways, coloured red, green and blue. However you tile it, each colour appears exactly n² times.

Try it

Solve builds a random tiling. Scramble throws the pieces out again. Build your own: click a lozenge to turn it, drag it to move it (it snaps to the grid). Count the colours when you're done.

The proof is a change of perspective

Look at a finished tiling and stop seeing it as flat: it is a drawing of unit cubes stacked in the corner of an n × n × n box. The three colours are the three kinds of visible face: tops, lefts and rights.

Now press a colour. The pile is lifted into 3D and the camera turns to look straight down one axis of the box. From there only faces of that colour are visible, and together they cover one n × n wall of the box exactly. So there are n² of them. The same goes for the other two colours.

The perspective buttons need a complete tiling; press Solve first if you've been building by hand.

Where it comes from

This is the problem of the calissons, named after the lozenge-shaped sweets of Aix-en-Provence. Guy David and Carlos Tomei published the cube proof in the American Mathematical Monthly in 1989. It is one of the classic proofs without words.

Under the hood

A lozenge tiling is the same thing as a perfect matching between the hexagon's up-triangles and down-triangles. Solve finds a random one with Hopcroft–Karp. The 3D view assigns each vertex a height from the tiling and rotates the camera.